In triangle ABC, point D lies on CA and point E lies on CB. Angle CDE is congruent to angle CBA, but DE is not parallel to AB. If CD = 4, CB = 10, and CE = 5, what is the length of CA?

SAT Lines, angles, and triangles Practice Question 7 (Hard)

Question bank / Lines, angles, and triangles

Math · Geometry and TrigonometryHard

In triangle $ABC$ , point $D$ lies on $\overline{CA}$ and point $E$ lies on $\overline{CB}$ . Angle $CDE$ is congruent to angle $CBA$ , but $\overline{DE}$ is not parallel to $\overline{AB}$ . If $CD = 4$ , $CB = 10$ , and $CE = 5$ , what is the length of $\overline{CA}$ ?

$8$

$9$

$12.5$

$15$

Show answer and explanation

Answer: C

Triangles $CDE$ and $CBA$ share angle $C$ , and angle $CDE \cong$ angle $CBA$ , so they are similar with $D$ corresponding to $B$ and $E$ corresponding to $A$ . Corresponding sides are proportional: $\frac{CD}{CB} = \frac{CE}{CA}$ . So $\frac{4}{10} = \frac{5}{CA}$ , which gives $CA = 12.5$ .

Choice A comes from matching $D$ with $A$ as if $\overline{DE} \parallel \overline{AB}$ ( $\frac{4}{CA} = \frac{5}{10}$ ), but the given congruent angles pair $D$ with $B$ .

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